Stationary critical points of the heat flow in spaces of constant curvature

The paper considers stationary critical points of the heat flow in sphere S^N and in hyperbolic space H^N, and proves several results corresponding to those in Euclidean space R^N which have been proved by Magnanini and Sakaguchi. To be precise, it is shown that a solution u of the heat equation has a stationary critical point, if and only if u satisfies some balance law with respect to the point for any time. In Cauchy problems for the heat equation, it is shown that the solution u has a stationary critical point if and only if the initial data satisfies the balance law with respect to the point. Furthermore, one point, say X₀, is fixed and initial-boundary value problems are considered for the heat equation on bounded domains containing X₀. It is shown that for any initial data satisfying the balance law with respect to X₀ (or being centrosymmetric with respect to X₀) the corresponding solution always has X₀ as a stationary critical point, if and only if the domain is a geodesic ball centred at X₀ (or is centrosymmetric with respect to X₀, respectively).